THE  VISCOSITY  COEFFICIENT  OF  AIR 


WITH    AN    INQUIRY    INTO    THE    EFFECT    OF   THE 
RONTGEN    RAY    THEREON 


BY  FREDERICK  G.  REYNOLDS 


SUBMITTED    IN    PARTIAL    FULFILLMENT    OF     THE     REQUIREMENTS 
FOR   THE    DEGREE    OF    DOCTOR    OF    SCIENCE    IN   THE 
FACULTY  OF  SCIENCE,  NEW  YORK  UNIVERSITY 


Salem 
TME  SALEM  PRESS  Co.,  SALBAV,  MASS. 

1904 


THE  VISCOSITY  COEFFICIENT  OF  AIR 


WITH    AN    INQUIRY    INTO    THE    EFFECT    OF    THE 
RONTGEN    RAY    THEREON 


BY  FREDERICK  G.  REYNOLDS 


SUBMITTED    IN    PARTIAL    FULFILLMENT    OF     THE     REQUIREMENTS 
FOR   THE    DEGREE    OF    DOCTOR   OF    SCIENCE    IN   THE 
FACULTY  OF  SCIENCE,  NEW  YORK  UNIVERSITY 


Salem  press: 

THE  SALEM  PRESS  Co.,  SALEM.  MASS. 
1904 


-: :  '.  •"•/I"  •    :  ;\  '  * 
•* : ••" *  •••    • :« •*•  •*"*•  "•" 


THE  VISCOSITY  COEFFICIENT  OF  AIR 

WITH    AN    INQUIRY    INTO    THE    EFFECT    OF    THE    RONTGEN 
RAY    THEREON 

BY  FREDERICK  G.  REYNOLDS 

ONE  of  the  chief  features  of  the  gaseous  form  of  matter 
is  the  distinguishing  simplicity  of  its  laws.  The  molecules  of 
the  medium  in  this  state  seem  to  act  in  a  less  complicated 
manner  than  when  in  a  liquid  or  solid  state.  The  investigation 
of  any  property  of  matter  in  the  form  of  a  gas  might  therefore 
be  more  simple  than  a  similar  investigation  on  any  other  form 
of  matter,  and  the  results  attending  such  an  effort  might  serve 
to  throw  some  light  on  these  other  and  more  complicated  forms. 

The  particular  phenomenon  in  question  has  claimed  the 
attention  of  investigators  for  many  years  and  while  the  general 
truths  may  be  said  to  be  settled,  there  have  always  existed 
many  differences  of  detail  in  which  one  may  find  a  justification 
for  further  effort  in  the  same  field. 

By  the  viscosity  of  a  gas  is  meant  the  resistance  which  it 
offers  to  a  continuous  change  of  form  depending  on  the  rate  at 
which  that  change  is  effected.  The  coefficient  of  viscosity 
may  perhaps  be  best  understood  by  considering  a  stratum  of 
the  gas  infinite  in  extent.  If  this  stratum  were  enclosed  by 
two  parallel  planes  at  a  distance  d  centimetres  from  each  other 
and  if  one  of  these  planes  were  to  move  parallel  to  the  other 
with  a  velocity  of  v  centimetres  per  second,  keeping  always  in 
the  same  direction,  till  the  gas  embraced  takes  up  its  final 
velocity  thus  imparted,  this  velocity  would  decrease  uniformly 
as  we  pass  from  the  moving  plane  towards  the  other.  If  again 
the  strata  in  contact  with  the  two  planes  have  the  same  velocity 
as  the  planes  themselves,  the  strata  will  decrease  in  velocity 

«  centimetres  per  second  for  every  centimetre  we  depart  from 
the  moving  plane.     The  friction  between  any  two  contiguous 

(3) 


REYNOLDS 

strata  will  then  be  the  same  as  between  either  plane  and  the 
stratum  in  contact  with  it.  If  this  friction  be  denoted  by  a 
tangential  force  F  on  every  square  centimetre  of  surface  then 

F  =  U  -3  where  U  is  the  coefficient  of  viscosity  of  the  gas. 

The  measurement  of  the  internal  friction  or  viscosity  of  a 
gas  is  the  determination  of  the  value  of  this  coefficient. 

The  property  which  enables  one  layer  to  drag  along 
another  might  in  liquids  be  looked  upon  as  a  cohesion  acting 
in  opposition  to  the  motion  but  in  the  case  of  a  gas  we  must- 
look  for  another  explanation.  In  the  light  of  the  Kinetic 
Theory  we  look  upon  the  gas  as  manifesting  itself  by  the  very 
rapid  motion  of  its  molecules.  When  two  layers  are  moving 
therefore  with  unequal  velocities,  the  molecules  from  the  faster 
moving  layer  penetrate  the  slower  moving  one  with  their  faster 
forward  velocity  and  vice  versa,  so  a  measure  of  the  friction 
would  be  the  momentum  which  is  thus  carried  over.  The 
heat  motion  of  the  molecules  is  so  great  (being  at  0°  Cent,  a  mean 
velocity  of  447  metres  per  second),  that  it  is  not  affected  by 
the  comparatively  slow  forward  motion  of  the  molecules.  It 
would  naturally  follow  that  this  friction  would  increase  with  an 
increase  of  the  temperature  for  the  speed  of  the  molecules 
increases  with  the  temperature,  being  proportional  to  the  square 
root  of  the  absolute  temperature.  That  the  friction  is  inde- 
pendent of  the  density  seems  paradoxical  at  first,  but  the 
theory  is  at  least  justified  when  we  consider  that  the  transfer 
is  made  by  the  molecules  and  is  greater  as  their  number  and 
activity  are  greater,  hence  as  the  density  is  greater ;  but  at  the 
same  time  the  transfer  takes  place  in  layers  whose  distance 
apart  may  be  traversed  by  the  molecules  and  as  the  density 
increases,  this  distance  is  restricted  so  that  it  is  possible  that 
the  combined  result  of  the  two  causes  should  produce  a  con- 
stant effect. 

Internal  Friction  as  a  property  of  fluid  media  is  spoken  of 
as  early  as  the  year  1687  by  Newton1  who  makes  use  of  the 

1  Phil.  Nat.  Princ.  Math.  1687,  Lib.  II,  Sect.  9. 


THE    VISCOSITY    COEFFICIENT    OF    AIR  5 

following  language :  "  Attritus  vel  resistentia  quee  oritur  ex 
defectu  lubricitatis,"  and  he  advanced  the  theory  that  the 
friction  between  two  neighboring  layers  of  a  fluid  does  not 
depend  upon  the  pressure,  that  it  is  proportional  to  the 
difference  of  the  velocities  of  the  layers  and  that  it  is  propor- 
tional to  the  size  of"  the  surface. 

The  methods  of  approaching  and  handling  the  investiga- 
tions into  the  viscosity  of  fluids,  both  theoretical  and  practical, 
may  be  divided  into  four  general  classes  :— 

I.  By  their  transpiration  through  capillary  tubes,  in 
which  phase  of  the  investigation  Poiseuille1  and 
Graham2  were  the  pioneers,  the  former  as  to  liquids 
and  the  latter  as  to  gases. 

II.  By  the  swinging  of  pendulums,  a  method  followed 
by  Baily3,  Bessel4  and  Du  Buats  all  of  which  is  dis- 
cussed in  a  paper  by  Stokes6  "  On  the  Internal 
Friction  of  Fluids  on  the  Motion  of  Pendulums." 

III.  By  the  torsional  vibrations  of  an  immersed  disc,  first 

adopted  by  Coulomb.7 

IV.  By  the  torsional  vibrations  of  a  vessel  filled  with  the 

fluid  in  question  which  might  rather  be  considered 
a  development  of  the  preceding  method.8 
Each  method  seems  to  have  its  practical  and  theoretical 
advantages  and  disadvantages  and  the  discordant  results  at- 
tending the    early  efforts    in    each   have  since    been  brought 
more  closely  into  harmony  through  criticism  and  refinements 
of  process.       In    speaking  of  the  transpiration  and    torsion 
methods  Meyer  remarks  of  the  discrepancies  "als  begrundet  in 
den  Annahmen,  welche  zur  Gewinnung  brauchbarer  Formeln 

1  Mem.  de  Savants  Strangers,  1846,  IX,  p.  433. 

2  Phil.  Trans.  1846,  CXXXVI,  p.  573;   1849,  CXXXIX,  p.  349. 
a  Phil.  Trans.  1832. 

4  Berlin  Acacl.  1826. 

5  Principes  d'Hydraulique,  1786. 

6  Cambridge  Phil.  Trans.  1850,  IX,  pt.  2. 

7  Mem.  de  1'Institut  National,  III,  p.  246. 

s  Helmholtz  and  Pietrowski,   Sitzungsber.   der  kk.    Akad.    April,  1860. 


6  REYNOLDS 

in  die  theoretische  Entwickelung  eingefiihrt  wiirden  ;  die  dabei 
eintretenden  Vernachlassigungen  waren  in  dem  Falle  schwin- 
gender  Scheiben  derart,  dass  der  Coefficient  zu  gross,  in  dem 
Falle  des  Ausflusses  durch  Capillarrohren  derart  dass  erzu  klein 
erhalten  werden  miisste ;  die  wahren  Werthe  wiirden  also 
zwischen  den  Resultaten  diesen  beiden  -Methoden  liegen." 
Which  method  comes  nearer  to  the  truth  is  not  decided,  but 
the  possible  objection  to  the  transpiration  experiments  lies  in 
the  difficulty  of  measurement  of  the  diameters  of  the  tubes  and 
also  in  the  fact  that  on  account  of  the  smallness  of  the  bore  one 
cannot  be  certain  that  the  action  between  the  molecules  of  the 
gas  and  of  the  substance  of  the  tube  does  not  affect  the  result. 
The  pendulum  method  seems  to  be  considered  capable  of  great 
accuracy,  but  in  the  theory  applicable  toitBaily  made  the  error 
of  considering  that  the  viscosity  is  dependent  on  the  density. 
A  correction  of  his  results  on  the  basis  of  a  correct  hypothesis 
has  been  promised  but  does  not  seem  to  have  been  made.  The 
difficulties  or  objections  connected  with  the  method  of  a  vibra- 
ting disc  seem  to  be  less  than  those  in  the  transpiration  method 
and  it  may  be  for  this  reason  that  it  has  been  the  method  gener- 
ally selected  by  the  later  investigators  ;  but  the  trouble  here  lies 
in  the  necessity  of  accounting  properly  for  the  effect  of  the 
motion  of  the  medium  near  the  edge  of  the  discs.  As  a  result 
of  Maxwell's1  analysis  and  deductions  given  in  1860,  both  he 
and  Meyer2  started  to  demonstrate  practically  the  truth  of  the 
same,  using  as  a  basis  of  their  investigations  the  method  adopted 
by  Coulomb  ;  but  Meyer  altered  the  earlier  method  in  so  far  as 
to  substitute  three  oscillating  discs  with  a  common  axis  in- 
stead of  the  one  used  by  Coulomb.  These  plates  were  so  ar- 
ranged that  it  was  possible  to  separate  them,  thereby  exposing 
six  surfaces  or  to  place  them  together  exposing  only  the  two. 
In  this  way  the  friction  on  the  exposed  parts  of  the  hanging 
system  such  as  the  mirror,  etc.,  as  well  as  the  internal  friction 
of  the  wire  itself,  could  be  eliminated  by  considering  the  differ- 

1  Phil.  Mag.  1860,  (4)  XIX,  p.  31;  Phil.  Trans.  1866,  CLVI,  p.  249. 

2  Pogg.  Ann.  1865,  CXXV,  p,  177. 


THE    VISCOSITY    COEFFICIENT    OF    AIR  7 

ence  of  the  effects  obtained  when  the  plates  were  apart  and  to- 
gether. Maxwell  improved  on  this  by  inserting  four  fixed 
plates  between  the  moving  ones,  which  had  the  effect  of  caus- 
ing the  friction  of  the  gas  to  exert  a  greater  resistance  and 
therefore  one  more  easily  measured,  an  advantage  which  was 
immediately  recognized  by  Meyer1  himself.  A  further  ad- 
vantage of  Maxwell's  method  lies  in  the  fact  that  his  deductions 
give  the  viscosity  coefficient  directly,  whereas  Meyer's  deduc- 
tions lead  to  the  square  -root  which  would  cause  the  effect  of 
any  error  to  be  increased.  As  a  consequence  Maxwell's  results 
showed  a  better  agreement  in  the  effort  to  verify  the  independ- 
ence of  the  coefficient  and  the  pressure  within  the  range  of  his 
observations,  viz.  :  from  one  to  ^L  atmospheres.  In  1887,  as  a 
result  of  a  correction  suggested  by  Koenig,2  an  improvement 
was  made  by  Meyer  in  his  method  and  he  calculated  his  results 
again  on  the  improved  theory3. 

This  independence  of  the  viscosity  and  the  pressure  cannot 
be  expected  to  hold  in  the  limiting  cases,  as  would  follow  from 
a  consideration  of  the  theory.  The  density  and  the  free  path 
enter  we  have  seen  as  factors,  but  when  the  former  approaches  0 
the  latter  approaches  co  which  would  cause  this  phenomenon  to 
fall  outside  of  a  development  where  only  finite  densities  and 
free  paths  are  considered. 

Investigations  on  this  point  at  low  pressures  were  taken  up 
by  Kundt  and  Warburg4,  who  followed  Maxwell's  methods  and 
found  that  below  a  pressure  of  -$\  atmospheres  there  was  a 
marked  diminution  in  the  logarithmic  decrement  from  which  the 
viscosity  coefficient  is  calculated.  They  however  found  an  ex- 
planation of  this  in  a  phenomenon  which  they  called  Slipping 
(or  External  Friction),  by  which  is  meant  the  sliding  of  the 
medium  along  the  bounding  surface  as  distinguished  from  the 

1  Pogg.  Ann.  1871,  CXLIII,  p.  14. 

2  Wied.  Ann.  1887,  XXXII,  p.  193. 

a  Wied.  Ann.  1887,  XXXII,  p.  642;  Sitzuugsber.  d.  Miinchener  Akad.  1887 
XVII,  p.  343. 

4  Monatsber.  d.  Berl.  Akad.  1875,  p.  160;  Pogg.  Ann.  1875,  CLV,  pp.  337, 
525. 


8  REYNOLDS 

sliding  of  one  layer  of  the  medium  along  another  layer ;  and 
they  explained  this  falling  oft'  as  not  caused  by  a  decrease  of 
the  coefficient  of  viscosity,  but  by  an  increase  of  the  Slipping 
at  the  lower  pressure.  Even  a  highly 'polished  surface  must,  in 
comparison  with  the  molecules  of  the  gas,  be  considered  rough, 
and  under  ordinary  pressures  a  sliding  along  this,  especially 
since  the  forward  motion  is  very  slow,  is  practically  impossible, 
but  this  would  not  be  the  case  in  a  very  rare  medium.  By  the 
term  Coefficient  of  Slip  they  designated  the  ratio  of  the  internal 
to  the  external  friction,  with  the  result  that,  between  the  pres- 
sures of  .ti  and  20  millimetres  of  mercury,  the  value  of  the 
Coefficient  of  Slip  was  found  to  be  inversely  proportional  to 
the  density  of  the  gas  and  very  nearly  equal  to  the  free  path  of 
the  molecules.  Therefore  the  external  friction  is  directly  pro- 
portional to  the  density.  Warburg1  experimented  further  on 
the  external  friction  by  the  transpiration  method  and  obtained 
smaller  values  for  the  Coefficient  of  Slip  but  Breitenbach2  later 
obtained  results  nearer  the  earlier  ones. 

The  correctness  of  this  explanation  of  Kundt  and  Warburg 
for  the  sudden  falling  off  of  the  logarithmic  decrement  was 
later  proven  by  Crookes3  on  a  theory  developed  by  Stokes,4 
wherein  he  used  his  vacuum  tubes  with  vertically  suspended 
strips  of  mica  thereby  excluding  the  possibility  of  any  external 
friction.  The  results  of  these  investigations  show  the  inde- 
pendence of  viscosity  and  pressure  down  to  pressures  so  low 
as  to  be  no  longer  accurately  measured.  However,  at  much 
higher  vacuums  there  is  a  break  in  the  constancy. 

At  extremely  high  pressures  the  law  does  not  hold,  but 
this  might  also  be  expected  from  the  assumptions  in  the  theory. 
Starting  with  the  hypothesis  that  the  particles  traverse  straight 
paths  between  their  successive  encounters  with  each  other,  the 
curved  paths  traversed  during  the  period  of  actual  encounter 
would  be  negligibly  small,  at  ordinary  densities  or  at  low 

1  Pogg.  Ann.  1876,  CLIX,  p.  399. 
»  Wied.  Ann.  1899,  LXVII,  p.  826. 

3  Phil.  Trans.  1881,  CLXXII,  p.  387. 

4  Phil.  Trans.  1881,  CLXXII,  p.  435. 


THE    VISCOSITY    COEFFICIENT    OF    AIR  9 

/ 

densities ;  but  this  would  not  be  the  case  when  the  density  be- 
comes very  great.  The  law  considers  the  molecules  unchange- 
able and  the  Dissociation  of  the  molecules  under  this  extreme 
condition  may  contribute  to  this  variation.  This  would  be 
particularly  true  in  the  consideration  of  vapors.  Warburg 
and  Babo1  have  shown  in  the  case  of  carbonic  acid  that  the 
coefficient  of  viscosity  increases  with  the  density  at  pressures 
from  30  to  120  atmospheres,  which  law  seems  to  hold  for  other 
gases  as  well. 

In  1866  Meyer2  published  a  theory  wherein,  assuming  the 
independence  of  pressure  upon  the  friction,  he  gave  a  law 
governing  the  speed  of  flow  of  gases  through  capillary  tubes 
which  corresponded  with  the  law  found  by  Poiseuille  for  liquids 
and  which,  by  considering  the  amount  of  gas  transpired  in  a 
given  time,  leads  to  the  value  of  the  coefficient  of  friction.  His 
calculations  from  the  results  of  Graham's  observations  proved 
the  correctness  of  his  assumption.  He  obtains  for  the  coeffi- 
cient of  viscosity  of  the  air  at  0°  Cent,  the  values  .000171, 
.000170,  .000174  and  he  also  gives  the  coefficients  for  nine- 
teen other  gases.  Obermayer,3  Puluj4  and  Mutel,  following 
this  method,  obtain  for  air  the  values  .000167,  .000180, 
.000172  respectively. 

As  to  the  fact  of  the  increase  of  the  viscosity  with  the  in- 
crease of  the  temperature  there  can  be  no  doubt,  but  as  to  the 
rate  of  such  change  there  is  not  such  an  agreement  among  the 
different  investigators.  Maxwell  first  put  it  as  increasing 
directly  as  the  absolute  temperature  or  proportional  to  (1  +  ad) 
where  6  is  the  temperature  from  the  freezing  point  and  a  is  the 
coefficient  of  expansion.  Later  experiments  show  that,  for  air 
at  least,  it  does  not  increase  so  rapidly.  Efforts  were  made  to 
represent  the  change  by  a  factor  of  the  form  (1  +  ad)11  where 
n  was  first  given  as  f  ;  but  later  Barus,5  adopting  the  transpiration 

1  Wied.  Ann.  1882,  XVII,  p.  390;  Sitz.  d.  Berl.  Akad.  1882,  p.  509. 

2  Pogg.  Ann.  1866,  CXXVII,  pp.  253,  353. 

3  Carl's  Rep.  1876,  XII,  p.  13. 

4  Wien.  Ber.  1874,  LXIX(2),  p.  287;  LXX(2),  p.  243. 

5  Bull,  of  thell.  S.  Geol.  Surv.  No.   54,   Washington  1889.     Wied.  Ann. 
1889,  XXXVI,  p.  358. 


10  REYNOLDS 

method  and  with  a  range  of  temperature  from  0°  to  1300°,  gave 
to  n  the  value  f  for  air  and  hydrogen.  Different  values  were 
found  by  others  for  other  gases  and  Wiedeman1  found  that  n 
became  smaller  with  the  increase  of  temperature,  a  fact  also 
found  by  Holman2.  Schumann3  however  chose  to  represent  the 
change  by  a  formula  consisting  of  two  factors  (1  +  7#)2y/l  +  aO 
from  the  consideration  that  the  viscosity  is  dependent  upon  two 
things,  the  molecular  free  path  and  the  molecular  speed;  and 
while  this  satisfied  his  own  observations  founded  upon  Maxwell's 
method  it  did  not  agree  with  those  of  Barus.  Sutherland,4  tak- 
ing into  consideration  the  apparent  sphere  of  action  of  the 
molecules  as  distinguished  from  the  real  sphere  of  action, 
changed  the  first  factor  and  represented  the  change  by  the 
equation,— 


. 

0  +  e) 


0  being  the  absolute  temperature,  a  the  coefficient  of  expansion 
and  c  a  constant  depending  on  the  attractive  forces  exerted  be- 
tween the  molecules  when  near  together.  With  this  he  found 
results  agreeing  well  with  Barns  and  Holman. 

Vapors  as  distinguished  from  gases  proper  have  also  been 
the  subject  of  observations  to  determine  if  they  follow  the  laws 
of  gases  as  to  viscosity.  In  these  researches  L.  Meyer5  and 
Koch6  have  followed  the  transpiration  method,  while  Puluj7  and 
Schumann8  have  used  the  oscillation  method  with  the  result  that, 
at  ordinary  pressures,  like  true  gases  the  coefficient  of  viscosity 
is  independent  of  the  pressure.  However  the  formulae  given  for 
true  gases  do  not  here  represent  exactly  the  change  with  the 

1  Arch.  d.  Sc.  Phys.  et  Nat.  1876,  LVI,  p.  273. 

2  Proc.  Amer.  Acacl.  Boston  1877,  XII,  p.  1;    Phil.    Mag.  (5),    III,  p.  81; 
XXI,  p.  199. 

3  Wied.  Ann.  1884,  XXIII,  p.  353. 

4  Phil.  Mag.   1893  (5)  202.  p.  507. 

5  Wied.  Ann.  1879,  VII,  p.  497;    1881,  XIII,  p.  1;    1882,  XVI,  pp.  369,  394. 

6  Wied.  Ann.  1883,  XIX.  p.  857. 

7  Wien.  Ber.  1878,  LXXVIII  (2),  p.  279;  Carl's  Repert.  1878,  XIV,  p.  573. 

8  Wied.  Ann.  1884,  XXIII,  p.  353. 


THE    VISCOSITY    COEFFICIENT    OF    AIR  11 

temperature  and  it  seems  that  the  viscosity  changes  in  a  much 
larger  ratio.  Attempts  to  justify  this  disagreement  have  been 
made  wherein  the  causes  which  make  a  perfect  gas  differ  from 
the  laws  of  vapors  are  used  for  a  basis  of  argument.  Of  these 
perhaps  the  principal  one  which  might  be  mentioned  is  the 
phenomenon  of  Dissociation. 

The  question  of  the  effect  of  Electricity  and  Magnetism  on 
the  viscosity  of  a  gas  has  not  as  yet  been  very  satisfactorily  an- 
swered, but  that  these  cause  no  effect  would  appear  from  the 
results  of  Pagliani1  and  Noack2.  Koenig3  oscillated  crystal 
spheres  in  a  magnetic  field,  but  the  effect  on  the  coefficient  of 
viscosity  was  not  apparently  considered.  Quincke4  also  inves- 
tigated the  viscosity  of  fluids  in  the  electric  field,  showing  an 
increase  without  any  apparent  change  in  the  period  of  swing  of 
the  spheres.  To  these  results,  however,  objection  was  made  by 
Boltzman5.  The  results  of  the  efforts  of  investigators  to  de- 
termine the  Coefficient  of  Viscosity  of  the  Air,  with  the  methods 
used  by  them  may  be  seen  by  the  table  on  page  12. 

As  a  consequence  of  the  corrections  in  theory  and  through 
the  refinements  of  process  the  results  of  the  different  methods, 
while  they  varied  widely  at  the  start,  have  been  brought  into 
closer  harmony.  Since  no  new  method  differing  in  principle 
from  these  has  been  given,  the  only  available  way  of  testing 
the  correctness  of  the  results  is  in  a  variation  of  the  methods 
at  hand.  The  method  of  torsional  vibrations  seems  to  permit 
of  a  wider  range  of  variation  and  therefore  to  be  best  suited 
to  such  a  purpose.  The  objection  to  the  use  of  discs,  however, 
has  been  mentioned  as  lying  in  the  allowance  to  be  made  for 
the  action  at  the  edge  of  the  disc ;  but  the  improvement  of 
method  in  substituting  several  in  place  of  the  one  partly 
obviated  this.  This  objection  would  not  of  course  hold  if  a 
spherical  surface  were  substituted  and  for  this  reason  as  well 

1  Ac.  Torino,  1885,  XX,  p.  615;  1887,  XXII,  p.  1. 

2  Wied.  Ann.  1886,  XXVII,  p.  289. 

3  Wied.  Ann.  1887,  XXXI,  p.  273. 

4  Wied.  Ann.  1897,  LXII,  p.  1. 

5  Wied.  Ann.  1897,  LX,  p.  399. 


REYNOLDS 


as  that  its  symmetry  renders  possible  a  rigorous  theoretical 
development,  the  sphere  or  spherical  shell  seems  to  answer  the 
requirements  best. 


Investigator. 


Method. 


Coef.  C.G.S.    Temp. 


Stokes 
Meyer1 

1  1 

Baily's  Pendulum 
Oscillation 
Bessel's  Exper. 
Girault's     <  ' 

.000104 
.000353 
.000275 
.000384 

0 

44 

44                        44 

.000360 

18 

44  2 

44 

Oscillation  (2°) 

44 

.000333 
.000323 

8.3 
21.5 

44 

« 

.000366 

34.4 

44   3 

44 

Transpiration 

.000168 
.000174 

0 
0 

Maxwell4 

Oscillation 

.000200 

18 

Puluj5 
Von  Obermayer6 

44    7 

Transpiration 

4  I 

(  ( 

.000179 
.000171 
.000168 

0 
0 
0 

Schumann8 

Oscillation 

.000168 

0 

Schneebeli9 
Tomlinson10 

4  4 

Transpiration 
Oscillation  (cylinder) 

44 

.000171 
.000179 
.000176 

0 
12.02 
10.64 

4  4 

.      44 

.000177 

14.63 

4  4 

4  4 

.000178 

11.69 

4  4 

"      (spheres) 

,000176 

9.97 

DESCRIPTION    OF    APPARATUS 

A  vertical  glass  tube  92.71  cm.  long  and  6.85cm.  in  diam- 
eter has  its  lower  end  fitted  air  tight  into  a  brass  collar  7  cm. 

1  Crelle's  Jour.  59,  p.  229. 

2  Fogg.  Ann.  1871,  CXLIII,  p.  14. 

3  Pogg.  Ann.  1873,  CXLVIII,  pp.  37,  203. 

4  Phil.  Trans.  1866,  156,  p.  249. 

5  Wien,  Sitz.  Abth.  2,  1874,  LXIX,  p.  278.     Wien.  Sitz.  Abtb.  2,  1874,  LXX, 
p.  243. 

6  Carl's  Rep.  1876,  XII,  p.  15. 

7  Wien.  Sitz.  1875,  LXXI,  p.  281 ;  1876,  LXXIII,  p.  433. 

8  Wiecl.  Ann.  1884,  XXIII,  p.  353. 

9  Arch,  des  Sci.  Phys.  et  Nat.  Geneve,  1885  (3)  XIV,  p.  197. 
10  Phil.  Trans.  (2)  1886,  CLXXVII,  p.  768. 


THE    VISCOSITY    COEFFICIENT    OF    AIR 


13 


high  which  screws  into  a  brass  circular  bed-plate  28cm.  in 
diameter.  The  lower  surface  of  the  plate  is  ground  and 
polished  so  that  the  ground  edge  of  a  bell  jar  when  placed 
against  it  may  form  an  air-tight  connection.  The  upper  end 
of  the  glass  tube  is  capped  with  a  brass  cylinder  in  the  form  of 
an  inverted  cup  whose  upper  surface  is  3.49cm.  thick  and  is 
pierced  by  a  conical  shaped  opening  into  which  fits  the  cone  of 
a  torsion  head,  which  being  carefully  ground  is  capable  of  easy 
motion  but  at  the  same  time  is  air-tight.  The  brass  collar  at 
the  base  is  fitted  with  a  plane  glass  window  which  is  adjusted 
air-tight  over  a  part  of  the  glass  tube  which  has  been  cut  away. 
In  this  way  the  observer  may  see  clearly  within  that  part  of 
the  cylinder  since  the  rays  of  light  are  not  interfered  with  by 
the  cylindrical  surface  of  the  tube.  The  bed-plate  rests  upon 
a  tripod  fitted  with  adjusting  screws  in  its  legs  so  that  the  glass 
cylinder  may  be  made  vertical.  By  an  arrangement  of  clamps 
acting  as  a  wedge,  a  bell  jar  may  be  attached  to  the  tripod  and 
held  tightly  against  the  under  surface  of  the  bed-plate. 

In  the  first  and  second  sets  of  experiments  a  hollow  brass 
spherical  shell  was  used  and  careful  measurements  of  its 
diameter  were  made  for  the  purpose  of  determining  any 
variations  from  a  truly  spherical  form  with  the  result  that  the 
diameter  used  as  an  axis  was  found  to  be  very  slightly  smaller 
than  were  the  diameters  perpendicular  to  it,  but  the  difference 
was  so  small  as  to  cause  no  appreciable  error  in  considering  the 
surface  spherical.  The  dimensions  were  therefore  taken  as 
follows:  external  diameter  12.68cm.,  weight  235.7  grammes 
and  moment  of  inertia  (calculated)  6316.0647.  This  sphere 
hung  within  the  bell  jar  and  was  attached  to  the  suspending 
wire  by  means  of  a  brass  rod  28.8  centimetres  long  whose 
lower  end  screwed  into  the  sphere  and  whose  upper  end  was 
fitted  with  a  cross  pin  in  the  shape  of  a  T  which  fitted  into  a 
hook-shaped  saddle  attached  to  the  wire.  The  wire  was  of 
German  silver  .0254  cm.  in  diameter  and  79.3cm.  long  and  was 
used  throughout  the  entire  set  of  experiments.  A  small  plane 
mirror  was  attached  to  the  brass  connecting  rod,  and  this  was 


14  REYNOLDS 

capable  of  a  perpendicular  adjustment  so  that  the  mirror  could 
be  brought  directly  opposite  the  opening  in  the  base  of  the 
apparatus.  The  whole  suspended  apparatus  was  thus  protected 
from  any  extraneous  currents  of  air  and  a  very  small  turn  to 
the  torsion  head  would  suffice  to  set  the  system  into  motion 
which  could  be  seen  through  the  agency  of  the  mirror.  The 
readings  were  made  by  means  of  a  tangent  scale  and  telescope 
directly  in  front  of  the  opening  opposite  the  mirror  and  from 
95  cm.  to  100  cm.  distant  therefrom.  The  temperature  of  the 
gas  was  observed  from  a  thermometer  placed  within  the  bell 
jar  and  easily  read  from  without.  The  length  of  time  taken 
for  a  swing  from  one  position  of  rest  to  the  next  was  obtained 
by  means  of  a  break-second  chronometer  with  an  electrical 
attachment  to  a  recording  apparatus  from  which  yj^  seconds 
could  be  estimated.  The  barometer  from  which  the  readings 
were  taken  hung  near  the  apparatus. 

A  preliminary  set  of  experiments  was  made  and  the  results 
were  calculated  for  the  purpose  of  determining : 

1.  The  best  working  limits  of  the  apparatus  as  used. 

2.  Whether   either   the    logarithmic    decrement   or  the 
period  of  one  swing  depends  upon  the  amplitude  of  the  same. 

3.  Whether  within  the  range  of  future  experiments  there 
is  any  change  corresponding  to  the  varying  barometric  pressure. 

4.  Any  peculiarities  liable  to  occur  and  therefore  to  be 
avoided. 

PRELIMINARY    SET 

The  method  of  proceeding  in  this  instance  was  as  follows  : 
The  sphere  was  first  attached  to  the  brass  rod  and  the  latter 
was  then  placed  with  its  end  in  the  saddle ;  the  bell  jar  would 
then  be  brought  up  and  with  as  little  jar  as  possible  would  be 
clamped  into  position.  Then  by  menus  of  the  torsion  head, 
which  would  be  moved  first  in  one  direction  and  then  in  the 
reverse,  an  oscillating  movement  would  be  given  to  the  sphere. 
In  many  cases  a  slight  pendulum  motion  would  also  be  given, 
but  with  a  little  practice  it  was  found  that  this  could  be  entirely 
stopped  by  a  slight  pressure  of  the  hand  on  the  outside  of  the 


THE    VISCOSITY    COEFFICIENT    OF    AIR  15 

apparatus.  The  oscillations  were  allowed  to  continue  some- 
times at  the  start  for  an  hour,  whereby  over  two  hundred  would 
have  occurred,  the  object  of  this  being  to  accustom  the  sus- 
pending wire  to  its  load  and  to  the  torsional  vibrations  so  that 
when  the  readings  proper  were  taken  the  damping  effect  of  the 
internal  friction  of  the  wire  itself  would  be  reduced  to  as  small 
and  as  constant  a  quantity  as  possible.  The  necessity  of  this 
precaution  was  shown  by  the  work  of  Tomlinson1  where  he 
found  that  rest  and  change  of  temperature  as  well  as  any 
sudden  shock  would  raise  temporarily  the  internal  friction  of 
the  wire.  After  this  the  sphere  would  be  given  a  fresh  swing 
and  when  the  amplitude  of  the'swing  was  the  desired  one,  the 
temperature,  barometric  pressure  and  time  of  start  would  be 
noted  as  well  as  the  starting  points  on  the  scale.  The  readings 
on  the  scale  were  always  taken  in  groups  of  ten  successive 
swings  and  at  the  end  of  every  two  or  three  groups  a  pressure 
on  the  electrical  key  would  record  the  time.  This  would  con- 
tinue till  two  hundred  readings  had  been  taken.  The  method 
of  getting  from  this  the  logarithmic  decrement  is  as  follows : 
Suppose  %,  a.2,  a3,  a4,  a5,  a6  and  61?  52,  53,  £>4,  b5  are  eleven  con- 
secutive readings  from  the  left  and  right  of  the  scale.  The  ten 
corresponding  arcs  from  rest  to  rest  are  therefore  al  +  b^  b±  +  a2, 
«2  +  62,  b.2  +  as,  «3  +  63,  63  +  a4,  a4  +  &4»  ^  +  «5,  a5  -f  65, 
65  +  a6.  The  mean  of  the  first  and  last,  second  and  ninth, 
third  and  eighth,  etc.,  would  then  be  taken  and  if  these  did  not 
vary  much,  as  was  generally  the  case,  the  mean  of  these  five 
would  be  taken  and  the  logarithm  of  the  corresponding  number 
of  seconds  noted.  This  would  also  be  done  with  the  second 
and  succeeding  sets  of  ten  readings  and  the  mean  of  the  ten 
differences  between  the  first  and  eleventh,  second  and  twelfth, 
etc.,  logarithms  would  when  divided  by  100  be  taken  as  the 
logarithmic  decrement.  These  differences  would  vary  slightly 
but  no  particular  law  of  variation  was  noticeable,  the  variation 
being  sometimes  in  one  direction  and  sometimes  in  the  other. 
The  accompanying  set  shows  more  particularly  the  method : 

1  "The  Influence  of  Stress  and   Strain  on  the  Physical  Properties    of 
Matter."    Phil.  Trans.  1886(2)  CLXXVII,  p.  801. 


16 


REYNOLDS 


log™  A 


n\ 


110.33 

22.666 

4.355375 

103.65 

21.303 

4.328441 

97.40 

20.027 

4.301616 

91.60 

18.841 

4.275104 

86.19 

17.693 

4.247802 

81.12 

16.695 

4.222587 

76.33 

15.714 

4.196286 

71.85 

14.795 

4.170115 

67.62 

13.926 

4.143825 

63.73 

13.127 

4.118165 

59.96 

.   12.353 

4.091772 

.263603 

56.41 

11.623 

4.065318 

.263123 

53.06 

10.935 

4.038818 

.262798 

50.05 

10.314 

4.013426 

.261678 

47.06 

9.700 

3.986772 

.261030 

44.28 

9.127 

3.960328 

.262259 

41.75 

8.607 

3.934852 

.261434 

39.12 

8.065 

3.906604 

.263511 

36.81 

7.589 

3.880185 

.263640 

34.67 

7.148 

3.854185 

.263980 

In  which  /^represents  the  respective  means  of  ten  consecu- 
tive amplitudes  as  noted  on  the  scale. 

A  the  amplitudes  reduced  to  seconds. 

n\  a  multiple  of  the  logarithmic  decrement  found  by  sub- 
tracting 11  from  1,  12  from  2  etc. 

The  mean  of  the  ten  differences  n\  gives  .2627056  and 
therefore  the  average  logarithmic  decrement  for  one  swing  is 
taken  as  .002627056.  The  results  of  such  observations,  taken 
at  different  times  under  varied  conditions  of  barometric  pres- 
sure and  temperature,  are  shown  by  the  table  on  page  17,  from 
which  it  appears : 

1°  The  small  difference  of  barometric  pressure  has  no  effect 
on  the  logarithmic  decrement. 

2°  The  period  of  oscillation  is  unaffected  by  the  amplitude 
of  swing  or  by  the  temperature. 

3°  The  logarithmic  decrement  is  unaffected  by  the  ampli- 
tude of  swing. 


THE    VISCOSITY    COEFFICIENT    OF    AIR  17 


Temp.       Barom.       Period         Amplitude          Log.  Dec. 


12.65 

30.21 

15.98 

121.1  — 

36.9 

.00257493 

15.25 

30.2 

15.91 

129.5  — 

199.5 

.00260613 

16.40 

29.95 

16. 

113.7  — 

33.7 

.00262706 

16.67 

30.15 

15.97 

168.7  - 

50.2 

.00262837 

17.85 

30.08 

15.98 

164.4  — 

84.2 

.00263851 

18.15 

30.06 

15.99 

74.1  - 

38.1 

.00264170 

18.22 

29.81 

15.97 

200.5  — 

199.5 

.00264190 

18.30 

30.18 

15.94 

205.1  — 

60. 

.00265430 

4°  The  temperature  is  the  only  element  which  affects  the 
logarithmic  decrement.  This  was  further  emphasized  in  two 
cases  where  during  the  observation  the  temperature  changed  grad- 
ually to  the  extent  of  one  degree.  In  one  case,  where  the  tem- 
perature increased,  the  value  of  the  logarithmic  decrement 
increased  steadily  instead  of  varying  in  one  way  and  the  other 
around  the  mean  value ;  and  in  the  other  case,  where  the  tem- 
perature gradually  fell,  there  was  a  steady  falling  off  hi  the 
value  of  the  logarithmic  decrement. 

This  sensitiveness  to  the  changes  in  temperature  rendered 
it  advisable  in  the  future  observations  to  confine  them  to  a 
shorter  period  of  time  so  that  there  would  be  less  liability  that 
a  change  in  temperature  would  occur,  and  therefore  in  the  fol- 
lowing sets  the  observations  were  confined  to  100  complete 
vibrations. 

SECOND    SET 

With  the  swinging  apparatus  as  described  it  was  of  course 
impossible  to  know  how  large  a  part  of  the  logarithmic  decre- 
ment was  due  to  the  internal  friction  of  the  wire  itself  and  to 
the  action  of  the  air  on  the  mirror.  A  change  therefore  was 
necessary  so  that  this  effect  could  either  be  eliminated  or  ac- 
counted for.  This  was  accomplished  by  an  addition  to  the 
apparatus  as  before  described.  A  hollow  brass  tube  14.818  cm. 
long  was  pierced  by  a  hole  in  the  plane  of  its  mid-section  so  as 
to  permit  the  hanging  brass  rod  to  pass  through  it  and  to  which 
it  was  fastened  so  as  to  hang  8.6  cm.  from  the  lower  end. 


18  REYNOLDS 

Great  care  was  necessary  in  the  arrangement  of  this,  so  that 
when  finished  it  should  hang  in  a  perfectly  horizontal  position 
and  at  right  angles  to  the  suspending  rod  in  order  not  to  de- 
flect the  mirror.  Two  solid  brass  cylindrical  weights  were  now 
constructed  which  fitted  snugly  into  the  ends  of  the  hollow  cyl- 
inder and  each  weight  was  carefully  cut  down  till  it  was  half 
the  weight  of  the  spherical  shell  (i.  e.  117.85)  when  the  length 
of  each  was  found  to  be  4.842  cm.  The  moment  of  inertia  of 
the  spherical  shell  had  been  calculated  to  be  6316.0647  and  it 
was  desired  to  place  each  cylinder  at  such  a  distance  from  -the 
axis  as  to  have  its  moment  of  inertia  equal  to  3158.03235.  The 
moment  01  inertia  of  each  around  an  axis  through  its  centre  of 
inertia  is  229.9353  and  therefore  2928.09705  is  the  moment 
about  the  original  axis  of  the  whole  mass  considered  as  placed 
at  the  centre  of  inertia.  If  the  distance  of  the  centre  of 
inertia  from  the  axis  be  denoted  by  x,  then 

MX*  =  2928.09705 
from  which 

x  =  4.988  cm. 

and,  since  it  was  also  necessary  to  have  the  ends  of  the  weights 
flush  with  the  ends  of  the  enveloping  cylinder,  this  determined 
the  proper  length  of  the  cylinder  14.818  cm.  The  aim  in  this 
was  to  reproduce  the  weight  and  moment  of  inertia  of  the 
spherical  shell  in  the  weight  and  moment  of  inertia  of  the  two 
cylindrical  weights. 

The  method  of  procedure  in  this  set  was  as  follows : 
The  cylinders  would  be  removed  and  the  sphere  attached 
to  the  brass  rod  and  the  apparatus  then  left  for  several  hours, 
usually  over  night.  At  the  end  of  this  time,  by  means  of  the 
torsion  head,  the  sphere  would  be  oscillated  for  some  time  to 
accustom  the  wire  to  the  oscillations,  and  then  the  vibrations 
would  be  started  again  for  the  readings,  care  always  being 
taken  to  avoid  any  pendulum  m'otion.  In  this  case  100  com- 
plete swings  from  rest  to  rest  were  taken  and  the  time 
thermometer  and  barometer  noted  as  before.  The  sphere 
would  then  be  detached,  care  being  taken  not  to  jar  the  wire, 


THE    VISCOSITY    COEFFICIENT    OF    AIR  19 

and  the  cylinders  would  then  be  inserted  into  place,  the 
preparatory  oscillations  given  and  then  100  more  readings  from 
the  scale  with  temperature  and  pressure  would  be  noted.  The 
two  swinging  systems  being  the  same  in  every  respect  save 
that  of  the  extra  surface  exposed  by  the  spherical  shell,  the 
difference  between  the  decrements  when  the  sphere  was  swing- 
ing and  when  the  counter  weights  were  inserted  was  that  due 
to  the  action  of  the  spherical  surface  alone.  The  method  was 
varied  by  taking  alternately  the  sphere  reading  first  and  then  the 
readings  with  the  weights  first,  and  the  whole  extended  over  a 
range  of  eight  degrees  in  temperature  during  a  series  of 
twenty-seven  observations.  A  slight  discrepancy  in  the  period 
of  swing  was  noticed  when  the  sphere  was  attached  and  when 
the  cylinders  were  in  place  which  led  to  a  further  investigation 
to  reconcile  the  results.  The  cylinders  were  removed  and 
circular  paper  caps  were  attached  to  the  horizontal  hollow 
cylinder  and  the  time  and  decrement  noted,  and  compared  with 
the  time  and  decrement  when  the  caps  were  removed  exposing 
the  interior  of  the  cylinder.  With  the  use  of  the  caps  the  times 
were  found  to  agree  and  a  slight  correction  was  found  necessary 
to  the  logarithmic  decrement  of  the  sphere  as  found  originally. 
The  corrected  results  are  inserted  in  the  following  table  : 

Time  Temp.  \i-\2 


18.965 

16.58 

.002672 

18.99 

16.91 

.002679 

19.02 

17.56 

.002691 

18.98 

18.27 

.002717 

19.005 

19.63 

.002760 

18.955 

20.72 

.002808 

18.97 

21. 

.002819 

18.972 

21.78 

.002851 

18.977 

22.36 

.002873 

18.99 

23.29 

.002904 

19. 

24.61 

.002924 

20  REYNOLDS 

KirchofF  in  his  "Mechanik"1  derives  a  formula  involving 
the  coefficient  of  viscosity,  the  logarithmic  decrement,  the  time 
of  swing  and  certain  other  known  elements  as  follows  :— 


In  which  rj  —  coefficient  of  viscosity. 
ft    =  density  of  the  medium. 
R  —  radius  of  the  swinging  sphere. 
X    =  the  logarithmic  decrement. 
T  =  the  swinging  time  in  the  fluid. 
T0  =  the  swinging  time  free  from  influence  of 

friction. 
K  =  moment  of  inertia  of  the  system,  which  in 

the  case  of  a  spherical  shell  =  %Ma?. 
The  relation  between  the  observed  Tand  T0  is  given  by  :— 


Using  for  the  density  of  the  air  the  value  .001293  in  the 
above  formula  the  value  of  the  coefficient  of  viscosity  at  the 
temperature  20.72  wag  calculated  to  be  : 

.00018697 

THIRD    SERIES 

The  object  of  this  series  was  to  see  if,  with  an  entirely 
different  swinging  system,  results  could  be  obtained  which 
would  approximate  closely  the  results  obtained  with  the  spher- 
ical shell.  Such  agreement  if  obtained  would  justify  the  mathe- 
matical theories  giving  rise  to  the  formula  used. 

For  this  purpose  two  hollow  brass  cylinders  were  pro- 
cured and  after  careful  tests  were  found  to  depart  but  the 

i  Vierte  Aufl.  26  Vorlesung,  p.  383. 


THE    VISCOSITY    COEFFICIENT    OF    AIR  21 

slightest  from  exact  uniformity  of  measurement.     The  dimen- 
sions were  as  follows : 


Cylinder 

Length 

Weight 

Inside 
Diameter 

Outside 
Diameter 

A 

30.4cm. 

327.8  gr. 

5.00 

5.08 

B 

30.4  cm. 

316.3  gr. 

4.70 

4.78 

A  thin  circular  brass  cap  was  soldered  to  one  end  of  the 
smaller  cylinder  in  the  centre  of  which  was  fastened  a  perpen- 
dicular brass  rod  16  centimetres  long  so  that  it  coincided  ex- 
actly with  the  axis  of  the  cylinder.  In  a  manner  similar  to  the 
one  used  on  the  sphere,  this  rod  was  fitted  at  the  top  extremity 
with  a  cross-pin  to  rest  in  the  hanging  hook  on  the  wire.  The 
wire  was  the  same  one  as  used  before,  its  length  being  this 
time  80.1  cm.  By  means  of  two  small  felt  collars,  one  on  the 
lower  end  of  the  smaller  cylinder  and  the  other  on  the  upper  end 
of  the  larger  one,  the  larger  when  placed  over  the  smaller  was 
held  snugly  in  place  and  the  whole,  when  placed  on  the  sus- 
pending wire,  hung  in  a  perpendicular  position  with  the  axis  of 
the  two  cylinders  coinciding  with  the  wire.  This  telescopic 
arrangement  of  the  cylinders  permitted  the  outer  one  to  be  drawn 
out  over  the  inner  one  or  pushed  back  so  as  to  cover  it  and  in 
this  way  it  was  possible  to  eliminate  the  effect  of  the  friction  of 
the  wire  and  the  action  of  the  air  on  the  mirror.  Two  marks 
25.4  cm.  apart  were  made  on  the  inner  cylinder  and  the  exten- 
sion was  limited  to  this  amount.  In  the  same  manner  as  before 
the  readings  were  made  from  the  scale  and  the  temperature, 
time  and  pressure  were  noted.  The  observations  were  always 
made  in  pairs ;  one  set  with  the  cylinders  collapsed  and  the 
other  with  the  cylinders  extended  and  alternately,  one  set 
and  then  the  other  being  taken  first.  The  results  of  34  sets  with 
temperature  varying  about  six  degrees  were  of  a  very  satisfac- 
tory character.  The  time  of  swing  averaged  13-09  seconds, 
any  one  differing  from  this  by  less  than  .05  seconds. 


22  REYNOLDS 


Temp. 

Cylinders 
Extended 

Temp. 

Cylinders 
Collapsed 

21.4 

.001504 

21.25 

.000953 

22.11 

.001507 

22.73 

.000954 

22.75 

.001509 

23.35 

.000956 

23.6 

.001518 

24.3 

.000960 

24.35 

.001522 

25.52 

.000961 

25.21 

.001526 

25.61 

.000962 

25.95 

.001529 

25.7 

.000964 

26.38 

.001531 

26.73 

.000966 

27.1 

.001533 

27.4 

.000969 

The  differences  between  the  two  corresponding  logarith- 
mic decrements  vary  from  .000551  to  .000564  and  these  differ- 
ences were  taken  for  the  logarithmic  decrement  corresponding 
to  a  cylinder  of  the  internal  and  external  diameters  of  the 
smaller  cylinder  and  25.4  cm.  in  length. 

From  the  mathematical  deductions  of  Stokes  the  logarith- 
mic decrement  due  to  the  action  on  the  walls  of  the  cylinder 
will  be  given  by  : 

M/JLT 

v/2  •  0.375/-1 


-  v/2  •  0.4922/-»  -f  etc.) 
where 


In  which  A,  =  the  logarithmic  decrement. 

M  —  mass  of  air  confined  in  a  cylinder  of  same 
length  and  mean  radius  of  the  one  con- 
sidered. 

yu.  —  the  coefficient  of  viscosity. 

r  =  the  observed  time. 

I  =  the  moment  of  inertia  of  the  cylinder  con- 
sidered. 

p  —  the  density  of  the  air  considered  as  about 
half  saturated  with  moisture. 

a  =  the  mean  radius  of  cylinder. 


THE    VISCOSITY   COEFFICIENT    OF   AIR  23 

In  applying  the  formula  a  first  approximation  was  made 
for  the  value  of  p  and  this  used  in  a  second  approximation. 
A  third  was  found  unnecessary  and  the  value  of  //.  resulting 
from  an  observed  time  13.09  seconds,  a  logarithmic  decrement 
.000551  and  at  a  temperature  21.33  was  calculated  to  be  : 

.00018711 

This  value  agrees  very  closely  with  the  value  found  by  the 
preceding  method  and  both  agree  well  with  the  later  results  of 
other  investigators  as  may  be  seen  by  the  table. 

FOURTH    SERIES 

Attention  has  been  called  by  Thomson1  to  the  facility  with 
which  a  gas  can  be  changed  from  a  conductor  to  a  non-con- 
ductor by  the  application  and  removal  of  Rontgen  rays  ;  and 
in  particular  he  caused  the  gas  so  exposed  to  be  swept  past  a 
wire  charged  with  electricity  and  in  this  way  part  of  the  charge 
would  be  carried  to  an  electrometer.  When  the  ray  was  not 
acting  on  the  gas  no  charge  was  carried  over  by  the  gas.  The 
gas  it  was  found  loses  this  conductivity  when  a  current  of 
electricity  passes  through  it  and  also  when  forced  through  a 
plug  of  glass  wool,  which  latter  fact  would  indicate  that  the 
structure  in  virtue  of  which  the  gas  conducts  is  of  a  coarse 
character,  since  it  does  not  survive  the  passage  through  the  fine 
pores  of  the  glass  wool.  The  study  of  any  other  property  of 
a  gas  when  in  the  state  into  which  it  is  thrown  by  the  Rontgen 
ray  might  therefore  lead  to  further  interesting  and  valuable 
results.  The  object  of  this  phase  of  the  investigation  is  to 
ascertain  if  there  is  any  difference  in  the  coefficient  of  vis- 
cosity of  air  when  it  is  affected  by  the  influence  of  the  Rontgen 
ray. 

In  this  set  the  brass  spherical  shell  used  before  was  hung 
in  position  and  the  bulb  from  an  X-ray  apparatus  was  brought 

1  Phil.  Mag.  1896  (5),  208,  p.  392.    "Discharge  of  Electricity  through  Gases." 


24  REYNOLDS 

within  15  cm.  to  the  surface  of  the  sphere.  It  was  found  nec- 
essary to  have  the  coil  in  an  adjoining  room  in  order  to  keep 
the  floor  and  air  free  from  the  vibrations  of  the  apparatus. 
Furthermore  in  order  to  shield  the  air  surrounding  the  sphere 
from  the  electrostatic  influences  of  the  bulb,  the  latter  was  sur- 
rounded by  a  metal  gauze  screen  which  was  connected  to  the 
earth.  The  observations  were  taken  in  pairs,  alternately  with 
the  current  on  and  off"  and  attention  was  first  directed  to  the 
period  of  swing.  In  the  following  table  where  the  average  time 
of  ten  consecutive  swings  is  given,  at  different  temperatures 
and  barometric  pressures,  no  evidence  of  any  uniform  law  of 
"change  is  apparent. 


Temp.            Bar. 

Without 
Ray 

With  Ray 

Amplitude 

15                29.6 

15.985 

16. 

80. 

—  70. 

44                                          4  4 

15.97 

16. 

19. 

-13.5 

tl                                          it 

15.95 

16.94 

16.6              29.77 

15.98 

15.98 

35. 

—  30. 

17.75 

15.94 

15.96 

70. 

—  64. 

17.75 

15.965 

15.94 

64. 

—56. 

15.97 

15.96 

84. 

7—71.5 

At  a  later  time  further  sets  were  taken  with  a  slight  change 
in  the  swinging  apparatus  and  at  higher  temperatures  and  the 
logarithmic  decrement  as  well  as  the  time  were  noted  for  com- 
parison. The  method  of  procedure  consisted  in  taking  the 
readings  of  thirty  complete  oscillations  with  the  current  off",  then 
thirty  more  with  the  current  on,  and  a  final  thirty  with  the  cur- 
rent on  again  ;  the  interval  between  each  set  of  thirty  being  the 
same,  usually  one  complete  oscillation.  The  average  of  the 
first  and  third  sets  was  taken  and  compared  with  the  second  set 
with  the  following  results  : 


THE    VISCOSITY    COEFFICIENT    OF    AIR  25 


Time 
off 

Time 
on 

Current 
off 

Current 
on 

Amplitude 

15.68 

15.7 

.0025498 

.0025428 

112.  —  94.9 

15.7 

15.73 

.0025681 

.0025647 

56.  _  47.6 

15.7 

15.7 

.0026147 

.0026042 

38.4—  32.4 

.0026893 

.0026223 

84.7—71.5 

15.7 

15.73 

.0025472 

.0025471 

120.6  —  102.3 

.0026653 

.0026273 

151.3  —  126. 

15.7 

15.7 

.0025697 

.0025648 

100.8—  85.4 

15.7 

15.7 

.0025791 

.0025729 

70.6—  59.8 

In  this  set  the  periods  agree  very  closely  but  it  might  be 
noted  that  the  variation  favors  usually  a  slight  increase  in  the 
period  of  swing  when  the  air  is  affected  by  the  Rontgen  ray 
but  the  difference  is  so  small  that  this  might  easily  be  caused 
by  errors  in  observation.  In  the  case  of  the  logarithmic  decre- 
ment the  difference  is  also  very  slight  but  points  to  a  decrease 
when  the  gas  is  under  the  influence  of  the  Rontgen  ray.  The 
effect  furthermore  seems  to  be  an  increasing  one  within  the 
limits  of  the  time  given,  since,  in  striking  the  average  decrement 
for  the  gas  when  under  the  influence  of  the  ray,  the  final  terms 
were  less  and  the  first  terms  greater  than  this  average  as  given. 
During  each  set  the  current  was  kept  practically  uniform  but 
it  was  varied  in  the  different  sets  wherein  the  more  pronounced 
effect  accompanied  the  stronger  current. 


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(26) 


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28 


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